## The Ecliptic and Celestial Equatorial Planes

The ecliptic (the path of the earth about the sun) is said to be in the ecliptic plane, which therefore contains the sun's and the earth's mass centers (Fig. 1.1). One also speaks of the celestial equatorial plane, which is the plane parallel to the earth's equatorial plane and through the sun's mass center. The ecliptic and celestial equatorial planes intersect, as illustrated in the figure, in a line referred to as the equinox line (because, when the earth on its annual path crosses this...

## N

Figure 3.4 The six orbital elements a, e, i, 2, u> , tp. T, first point of Aries N, north AN, ascending node P, periapsis h, angular momentum vector. Figure 3.4 The six orbital elements a, e, i, 2, u> , tp. T, first point of Aries N, north AN, ascending node P, periapsis h, angular momentum vector. this case, or when i 0, one often uses the quantity o co + 2 as one of the orbital elements. In the important case of circular orbits in the equatorial plane, e i 0. In this case just two orbital...

## Info

To help distinguish the forest from the trees, equation numbers that are printed in bold font indicate that the equation is either important, likely to be referred to in a later chapter, or else the final result of a preceding development. Simple corollaries or special cases of a preceding equation are given the same number, but with a prime. As would seem appropriate for a textbook of this type, journal papers or monographs are referenced only when the text or illustrations are so close to the...

## The Euler Equations for Time Dependent Moments of Inertia

In analyzing the attitude control of spacecraft, it can usually be assumed that the inertial properties are time independent. Although the properties will change as propellants are expended, this change is usually very slow compared with the attitude control frequencies and can be neglected. The attitude control of such spacecraft can then be described by means of the classical Euler's equations for the rotational motion of rigid bodies. In some other, more rare, instances the temporal change...

## Passive Nutation Damping of Oblate Spacecraft

Spinning, inertially symmetric spacecraft that are oblate, after disturbances have subsided, will continue to precess at a constant nutation angle (Sect. 6.3.1). In this sense, they are stable. The need, however, is to reduce the nutation angle to near zero after a disturbance. A number of different passive nutation dampers are in use. The principle upon which they are based is the same The precessional motion of the spacecraft induces a motion of the damping element relative to the spacecraft,...

## Earth Thermal Radiation and Albedo

The earth's surface, atmosphere, and clouds radiate into space at infrared wavelength (as corresponds to their relatively low temperature). At its surface, the earth is approximately in thermal equilibrium between the power absorbed from the incident solar radiation and its thermal emission. By comparison, the flow from the earth's interior of heat, produced by the radioactive decay of uranium and its daughter products, is only a minor contributor. The earth's thermal radiation follows roughly...

## P3 PO co2r wr2vtl2 cotan y453

High discharge pressures of the propellant pumps are necessary for injecting the propellants into the thrust chamber with a high pressure difference so as to form a spray of small droplets conducive to thorough mixing of the propellants. Also, the thrust chamber itself needs to be at a high pressure so as to avoid overexpansion. On the other hand, the inlet pressures must be chosen by a compromise between the need to mitigate against cavitation and the need to reduce the weight of the tank...

## Orbits And Trajectories In An Inverse

3.1 Kepler Orbits and Trajectories 60 3.2 Position as a Function of Time 65 3.3* D'Alembert and Fourier-Bessel Series 66 3.4 Orbital Elements 68 3.5* Spacecraft Visibility above the Horizon 71 3.6* Satellite Observations and the and g Series 73 3.7 Special Orbits 75 3.8* Perturbations by Other Astronomical Bodies 81 3.9* Planetary Flyby and Gravity Assist 85 CHAPTER 4. CHEMICAL ROCKET PROPULSION 97 4.1 Configurations of Liquid-Propellant Chemical Rocket Motors 99 4.2 Configurations of...

## Reference Frames

A quantitative description of the motion of space vehicles, be they launch vehicles, upper stage vehicles, artificial satellites, or deep-space probes, requires one or several reference frames with respect to which the vehicle's position, velocity, and acceleration as functions of time are defined. Reference frames are also needed to describe the vehicle's attitude relative, for instance, to points on the earth, to the sun, or to guide stars. Similarly, the time derivatives of the vehicle's...

## Blackbody Thermal Absorption and Emission

A blackbody is defined as an idealized material that absorbs all radiation incident on it, irrespective of the wavelength and of the angle of incidence. Reflection and transmission are therefore absent. As will be discussed here, the absorption and emission properties of a blackbody can be derived from one another, so that no new definition of the blackbody will be needed to describe its emission. The expression blackbody which has historical roots is somewhat inaccurate, because, as is clear...

## O

Figure 4.36 Cross section of a solid-propellant grain. (b) The ambient pressure above which flow separation in the nozzle is likely to occur. Make use of the Summerfield criterion, Eq. (4.25). (4)* Consider a propellant pump designed to pump liquid oxygen (density 1.19 103 kg m3 at a mass flow rate of 35.0 kg s. The pump is of the centrifugal type, similar to the one shown in Fig. 4.20. The flow enters the pump in the axial direction, then flows radially outward, driven by the impeller blades....

## Reference Frames and Coordinate Axes

The following reference frames and coordinates are in frequent use 3 . They are arranged in order of increasing closeness to ideal inertial frames. (la) Earth-fixed geocentric equatorialframe As shown in Fig. 1.2, this is the frame of reference conventionally used in cartography. Longitude is measured eastward from the Greenwich meridian (the former site of the Royal Observatory in Greenwich, a suburb of London). The latitude is measured positive northward, negative southward, starting at the...

## Heliocentric Osculating Elements of the Planets for 1996

The elements listed here are referred to the mean ecliptic and equinox at the Julian date 2000.0. First row under each planet is for the Julian date 2 450120.5, the second row for 2 450 320.5. Values given for Earth are actually for the Earth-moon barycenter. Taken from The Astronomical Almanac for the Year 1996, U.S. Government Printing Office, Washington, DC and Her Majesty's Stationery Office, London.

## Example The Sounding Rocket

Sounding rockets are comparatively small vehicles, used most often for scientific purposes, particularly for the study of the earth's upper atmosphere and ionosphere. Sounding rockets therefore do not need to reach orbit. They are often launched along a near-vertical trajectory. We consider here a single-stage rocket, launched vertically. The mass flow rate m is assumed constant until the time when all propellant is exhausted (or until the propellant flow is deliberately terminated so as to...

## Solar Thermal Emission

The sun's thermal emission originates in the photosphere. Its temperature can be obtained from spectral measurements and is found to be about 5950 K. Absorption by the layers above the photosphere modifies the radiation. Nevertheless, as illustrated in Fig. 7.3, the sun's thermal radiation into space is well approximated by blackbody radiation, albeit at the somewhat Figure 7.3 Solar spectral irradiance at 1.0 AU compared with the blackbody effective temperature 5760 K) and with the blackbody...

## Fundamentals of Thermal Radiation

The reader may already be familiar with much of the material on the theory of radiative heat transfer that is presented in this section. Much more can be found in standard texts, such as in the book by Siegel and Howell 2 . We present some basic material here primarily to define the concepts and notation that will be needed later. Also, this section will provide the opportunity to discuss early some applications specific to space technology before describing the more elaborate methods that are...

## Propellant Tank Configuration

Calculations based on shifting equilibrium of isentropic, one-dimensional expansion. a 1000 psi 689.5 N cm2. b Rocket Propellant-1. c Monomethylhydrazine. d Inhibited red fuming nitric acid. e Unsymmetric dimethylhydrazine. From Rocketdyne Division of Rockwell International, 1992 2 . Calculations based on shifting equilibrium of isentropic, one-dimensional expansion. a 1000 psi 689.5 N cm2. b Rocket Propellant-1. c Monomethylhydrazine. d Inhibited red fuming nitric acid. e Unsymmetric...

## Thermal Emission Absorption and Reflection of Technically Important Surfaces

The thermal radiation properties of technically important surfaces can differ greatly from those of the ideal blackbody surface. In principle, they could be obtained by quantum mechanical calculations. In practice, however, such calculations would be impossibly difficult to carry out, or else, if approximated, would be very inaccurate. Therefore such radiation properties can be obtained only by direct measurements. Whereas Eqs. (7.1) to (7.7) are thermodynamically precise, no such claims can be...

## Lumped Parameter Model of a Spacecraft

The design of a new spacecraft requires extensive thermal calculations. Their purpose is to make certain that the specified temperature limitations of the various components are not exceeded at any time during the life of the spacecraft. Such calculations must be performed for the different orientations of the spacecraft relative to the sun as they may occur during normal operating conditions. The same types of calculations are also carried out for times when the spacecraft is eclipsed. During...

## Inertial Measurement Units

Inertiai measurement units consist of a platform together with gyros and accelerometers on it. The platform is usually attached to the vehicle by gimbals. Most frequently there are three gyros and three accelerometers that have mutually orthogonal sensing axes, two parallel to the reference plane of the platform and one perpendicular to it. In a frequently used arrangement, the output from the gyros is used to maintain the platform by servo controls in a constant orientation relative to...

## Orbits and Trajectories in an Inverse Square Field

The principal forces that determine the path of a spacecraft are normally gravitation and thrust. In comparison, forces such as atmospheric drag or solar radiation pressure are small and can often be neglected. In this chapter the motion of space vehicles will be considered when gravity is the only force present, that is, at times when the vehicle is coasting without thrust or other forces acting on it. In the vicinity of a planet or other astronomical body, the gravitational field is composed...

## Solar Radiation Pressure

Solar radiation can exert an appreciable force and torque on a spacecraft. For instance, at an altitude of500 km in the earth's atmosphere, the effects of solar radiation on an orbiting spacecraft are of the same order of magnitude as the atmospheric drag. At higher altitudes, the solar radiation pressure predominates. The radiation pressure decreases with the inverse square of the distance from the sun. (Solar radiation needs to be distinguished from the solar wind, which is the name given to...

## 1

Figure 6.11 Hypothetical arrangement, with redundancy, of 24 attitude control and station-keeping thrusters. xu x2, x3 principal axes of spacecraft. closed would probably cause catastrophic impairments. A typical redundant propellant feed system is illustrated in Fig. 4.32. An attitude control feed system that would command the propellant flow to be proportional to demand would result in extremely small valves and thrusters and, as a consequence, would not be reliable. The fluid passages would...

## Atmospheric Entry

Reentry into the earth's atmosphere or entry into a planetary atmosphere calls for an understanding of the aerodynamic forces and moments on the vehicle and of the heat transferred to it. The flight paths are determined by gravity, drag, and lift. The lift, and to a lesser extent the drag, can be modulated by changes of the angle of attack, either by small thrusters or by movable aerodynamic surfaces. Also, the trajectories do not need to be restricted to a plane by banking the vehicle, without...

## Thermal Control Devices

As mentioned in Sect. 7.2, passive means of spacecraft temperature control can be obtained by selecting for the exterior surfaces of the spacecraft coatings or tapes that have the appropriate absorptivity and emissivity. Surfaces that are interior to the spacecraft are also treated in this manner. Black paint is applied to electronic boxes and other devices when electric-to-thermal dissipation is significant. An example is the digital module shown in Fig. 7.14. Although some of the generated...

## Thrust

In defining the thrust of a rocket motor as a force acting on the vehicle, one needs a definition that characterizes, to the extent possible, the motor performance in isolation from the other parts of the vehicle. When operating in the vacuum of space, the thrust is independent of the exterior of the vehicle. However, when operating in the lower atmosphere there will unavoidably be some interaction between the rocket plume and the air flow at the base of the vehicle. As is discussed further in...

## Lamberts Theorem

Lambert (1728-1777) established the surprising fact that in a central inverse-square force field the flight time between two points (fixed, as before, in the nonrotating reference frame attached to the center of attraction) is proportional to the Kepler orbit period, with a proportionality constant that depends only on the straight line distance between the points and the sum of the distances of the points from the center of attraction. Figure 5.2 illustrates the theorem and indicates the...

## Thermal Conduction

Heat transfer among spacecraft components is in part by radiation and in part by conduction. (Convective heat transfer, as it applies to rocket motors, is discussed in Sect. 4.14.) The reader is likely to be already familiar with the fundamentals of thermal conduction. Much material is readily available in several standard texts, such as the one by Carslaw and Jaeger 7 , The present section therefore merely lists some basic relations and adds some comments that apply to spacecraft. For an...

## Rate and Integrating Gyroscopes

Most spacecraft are equipped with gyroscopes of several different types. In particular, gyroscopes can be used for the determination of the orientation in inertial space of the chosen spacecraft reference axes and for the corresponding rates of change. Modern gyroscopes, either of the mechanical type or laser-ring gyros, are the product of a long development and have reached unprecedented precision. A large body of technical literature exists, much of it applicable to space technology. The...

## Euler Angles and Transformations of Coordinates

To describe the orientation of a three-dimensional Cartesian coordinate system with respect to another such system, in general three angles are needed. These then can serve as generalized coordinates. Different choices are possible, but choices usually most convenient in applications to spinning rigid bodies, and therefore also to space vehicles, are the Euler angles (p, 0, jr. To define them, we consider the Cartesian coordinate systems (Xi, X2, X3) and (Xi, x2, x t) as illustrated in Fig....

## J J [x xf dm J m

I , h, h are called the principal moments of inertia. (A single subscript is used to distinguish them from the moments of inertia in the general case.) It follows easily that i I2 k2, I3 X3 (6.11) If two of the eigenvalues and therefore also two of the principal moments of inertia are equal, the eigenvector associated with the third eigenvalue determines one of the principal axes. The other two principal axes are perpendicular to this axis and to each other but otherwise can be chosen...

## Example The YoYo Despin Mechanism

Both in orbit and during orbit insertion, spacecraft and upper stage vehicles are frequently provided with spin to ensure attitude stability. At other times it may be necessary to terminate the spin permanently or at least temporarily, for example, to reorient the vehicle prior to a propellant burn. Despinn-ing can be accomplished by firing small retro-rockets mounted on the vehicle such as to produce a torque counter to the direction of spin. Alternatively, despin can be obtained by initiating...

## Gravity Gradient Effect

If, over the space occupied by a space vehicle, the external gravitational field were perfectly uniform, the resultant of the forces on the various parts of the vehicle would be simply the force obtained by placing the entire mass at the vehicle's center of mass. In most applications, the assumption that the gravitational field is uniform over the extent of the space vehicle is satisfied to very high precision. Exceptions, referred to as gravity gradient effects, occur only when the dimensions...

## Extended Solid Rocket Nozzles

Figure 4.9 Performance comparison of comparable bell, aerospike, and ideal nozzles in the earth's atmosphere (the ideal nozzle as in Figs. 4.6 and 4.8). stagnation pressure po to the ambient pressure pa. The data are for the same values of the effective ratio of the specific heats, y 1.25, and of the area ratio Aex A* 50 that were used to construct Figs. 4.6 and 4.8. The curve labeled ideal nozzle represents the case of one-dimensional flow of an ideal gas without losses. 4.6.1 Comparison of...

## Thrust Vector Control With Thrust Chamber

Figure 4.23 Nodes for pogo oscillation analysis. (Adapted from Oppenheim and Rubin 11 .) Figure 4.23 Nodes for pogo oscillation analysis. (Adapted from Oppenheim and Rubin 11 .) The analysis starts by defining as discrete nodes all the components that are likely to play a role (Fig. 4.23). These include the thrust chambers, propellant pumps, ducts, their junctions, bellows, tanks and their outlets, hydraulic accumulators, and the principal vehicle structural elements. Each node will receive...

## Reaction Wheels and Gimbaled Momentum Wheels

Among the devices for attitude control, other than thrusters, are reaction wheels (Fig. 6.13). The wheels are driven by electric motors in either direction and are capable of high rotational speeds. They are supported by Figure 6.13 Schematic of reaction wheels and magnetic desaturation solenoids, (a) Assembly of three orthogonal wheels and their desaturation coils (b) rodlike desaturation solenoid with ferrite core. L, angular momentum of spacecraft and wheels Bg, earth magnetic field. Figure...

## Sidereal Time

Similarly to UT and GMST, sidereal time (Latin sidus star) is based on astronomical determinations. It uses the vernal equinox as a reference point. A sidereal day is therefore the time interval between two successive passages of the vernal equinox across the observer's meridian. Sidereal time is internally consistent with the geocentric equatorial reference frame discussed in Sect. 1.1.2. Because of variations in the rate of rotation of the earth, sidereal time is not exactly uniform. However,...

## Jet Damping

Rocket motor gas, as it streams through the motor case and the nozzle, generally has a damping effect on spacecraft motions. The phenomenon is referred to as jet damping 9 , As in aircraft, it can dampen pitch and yaw Figure 6.17 Schematic of a spinning spacecraft with nutation damping wheel and horizon sensor (wheel axis parallel to xY principal axis). motions. More significant, however, is the jet damping of spinning vehicles, because it counteracts the precessional motion by decreasing the...

## Problems

(1) Consider the nozzle represented in Fig. 4.35. An axisymmetric bell nozzle with zero divergence angle and the dimensions (in mm) indicated in the figure is assumed. There is zero ambient pressure at the nozzle exit. The calculation can be based on the approximation of a steady, one-dimensional, isentropic flow of a thermally and calorically perfect gas with an average ratio of 1.30 of the specific heats and a gas constant of 330 m2 (s2 K). The stagnation temperature can be taken equal to the...

## W2wi v2 v 2 Vpi Vpi v[i bnplHvpi vy1

I 4(Vpi Vi)((Vpi Vi) - 1) 1 + b npi)2(Vpi - V)4 1 (3.57) Energy is therefore gained by the spacecraft if and only if > V . The result also shows that for a given velocity ratio and impact parameter, the gain (or loss) is larger for massive planets. Comparing, for instance, Earth (mean equatorial radius 6378 km, mean orbital velocity 29.78 km s, mass 5.976 1024 kg) and Mars (3402 km, 24.13 km s, 6.418 1023 kg) and assuming the spacecraft's heliocentric velocity to be one-half the planet's...

## Universal Time UT and Greenwich Mean Solar Time GMST

In astrodynamics, for both historical and practical reasons, Universal Time is most often used. Closely related to it by a mathematical formula is the Greenwich Mean Solar Time. These systems of timekeeping are derived from observations of the sun's crossing the Greenwich meridian at noon. A solar day is the time interval between two successive crossings by the sun of the observer's meridian. Primarily because of the eccentricity of the earth's orbit about the sun, the length of the solar day...

## Attitude Control

The need for close control of the attitude i.e., orientation in space relative to some frame to be defined in each case of spacecraft follows from such requirements as the need to point antennas and sensors toward the earth or other astronomical objects. The required accuracy greatly depends on the particular application. It can be relatively low for broadcasting satellites. In some scientific applications the required accuracy of the attitude control may be 1 arc minute or less. Attitude...